Let X = {1,2,3,4,5} and Y={ 1,3,5,7,9}. Which of the following (s) is not relations from X to Y?a)R1 = {( x , y ) | y = 2+x, x ε X , y ε Y}b)R2 = {(1,1), (2,1), (3,3), (4,3), (5, 5)}c)R3 = {(1,1), (1,3), (3,5), (3,7), (5,7)}d)R4= {(1,3), (2,5), (2,4), (7,9)}Correct answer is option 'D'. Can you explain this answer?

Question

Let X = {1,2,3,4,5} and Y={ 1,3,5,7,9}. Which of the following (s) is not relations from X to Y?a)R1 = {( x , y ) | y = 2+x, x &epsilon; X , y &epsilon; Y}b)R2 = {(1,1), (2,1), (3,3), (4,3), (5, 5)}c)R3 = {(1,1), (1,3), (3,5), (3,7), (5,7)}d)R4= {(1,3), (2,5), (2,4), (7,9)}Correct answer is option 'D'. Can you explain this answer?

Answer

R1 = {(x, y) | y = 2x, x ∈ X, y ∈ Y} is a relation from X to Y because it pairs each element in X with the corresponding element in Y such that y = 2x.

R2 = {(x, y) | y = x^2, x ∈ X, y ∈ Y} is a relation from X to Y because it pairs each element in X with the corresponding element in Y such that y = x^2.

R3 = {(x, y) | y = x - 1, x ∈ X, y ∈ Y} is not a relation from X to Y because it pairs each element in X with an element in Y that is not in the set Y. For example, when x = 5, y = 4, which is not in Y.

Therefore, R3 is not a relation from X to Y.

Answer

We are given that
X = { 1 ,2 ,3 ,4 ,5 }
and Y= { 1 ,3 ,5 ,7 ,9 }
Since, (7, 9) ε R₄ but (7, 9) ∉ X x Y therefore R₄ is not a relation from X to Y.

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